Correspondence Analysis, Multiple Correspondence Analysis, and Recent Developments

Introduction

The use of multiple-choice response formats is common in psychology and other fields of inquiry. This format offers severaladvantages. First, it provides respondents with a faster and less tedious response format in comparison to rating or rank-order question formats. Second, its use leads to higher survey completion rates while enabling the inclusion of a greater number of questions and/or response categories in a survey (Arimond and Elfessi, 2001; Dolničar and Leisch, 2001). Third, the use of multiple-choice question formats represents a simpler means of data collection/management thus reducing data-entry costs (Javalgi et al., 1992). Finally, multiple-choice response formats are highly flexible in the sense that other types of categorical data such as binary, frequency table, and sorting data can be regarded as special cases of this general format (e.g., Nishisato, 1994; Takane, 1980).

Correspondence analysis (CA) and multiple correspondence analysis (MCA) represent descriptive multivariate techniques for exploring the associations inherent to multiple-choice questions (Benzécri, 1973; Gifi, 1990; Greenacre, 1984; Lebart et al., 1984; Nishisato, 1980). The distinction between CA and MCA rests in the former's focus on inter-relationships between two multiple-choice questions whereas the latter emphasizes inter-relationships among more than two multiple-choice questions. The reader is referred to Nishisato (2007) for an extensive historical overview of CA and MCA.

Technically, CA and MCA are closely related to canonical correlation analysis (CCA; Hotelling, 1936) and multiple-set canonical correlation analysis (MCCA; [Page 244]Carroll, 1968; Horst, 1961; Meredith, 1964), respectively. CCA is used to describe interrelationships between two sets of ‘continuous’ variables whereas MCCA captures those among more than two sets of continuous variables. In CCA and MCCA, a series of linear combinations or weighted composites of each set of variables, called the canonical variates, are obtained in such a way that they are mutually orthogonal to each other within the same set of linear combinations while remaining maximally correlated with different set(s) of linear combinations. These correlations between the variates are termed canonical correlations.

CA and MCA aim to construct linear combinations of the ‘response categories’ of multiple-choice questions in the same way as in CCA and MCCA, respectively. Thus they treat a single response category of each multiple-choice question as one variable in each set of variables in CCA and MCCA. CA and MCA typically display the weights for the linear combinations of response categories jointly in a low-dimensional graphical map. By representing inter-relationships among the response categories of multiple-choice questions in the map, CA and MCA have proved useful to both practitioners and academics alike (Hoffman et al., 1994). Moreover, they are non-parametric approaches and therefore do not require the a priori and correct specification of the distribution underlying multiple-choice data. Thus, CA and MCA are popular mapping methods that describe the association structures in multiple-choice data without recourse to stringent distribution assumptions (Green et al., 1987).

The purpose of this chapter is to provide an account of the technical underpinnings and applications of CA and MCA. As stated earlier, when data are in the form of multiple-choice questions, CA and MCA may be regarded as special cases of CCA and MCCA, respectively. Hence, we will begin with descriptions of CCA and MCCA so as to facilitate understanding of CA and MCA. Subsequently, we shall discuss two latest extensions of MCA – regularized MCA and a combined approach to MCA and a hard-clustering technique (c-means) for accommodating cluster-level respondent heterogeneity.

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